Transcript Slides for Rosen, 5th edition

Topic #1 – Propositional Logic
Module #1 - Logic
Propositional Logic (§1.1)
Propositional Logic is the logic of compound
statements built from simpler statements
using so-called Boolean connectives.
Some applications in computer science:
• Design of digital electronic circuits.
• Expressing conditions in programs.
• Queries to databases & search engines.
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George Boole
(1815-1864)
Chrysippus of Soli
(ca. 281 B.C. – 205 B.C.)1
Topic #1 – Propositional Logic
Module #1 - Logic
Definition of a Proposition
Definition: A proposition (denoted p, q, r, …) is simply:
• a statement (i.e., a declarative sentence)
– with some definite meaning, (not vague or ambiguous)
• having a truth value that’s either true (T) or false (F)
– it is never both, neither, or somewhere “in between!”
• However, you might not know the actual truth value,
• and, the value might depend on the situation or context.
• Later, we will study probability theory, in which we assign degrees
of certainty (“between” T and F) to propositions.
– But for now: think True/False only!
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Topic #1 – Propositional Logic
Module #1 - Logic
Examples of Propositions
• “It is raining.” (In a given situation.)
• “Beijing is the capital of China.” • “1 + 2 = 3”
But, the following are NOT propositions:
• “Who’s there?” (interrogative, question)
• “La la la la la.” (meaningless interjection)
• “Just do it!” (imperative, command)
• “Yeah, I sorta dunno, whatever...” (vague)
• “1 + 2” (expression with a non-true/false value)
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Operators / Connectives
An operator or connective combines one or
more operand expressions into a larger
expression. (E.g., “+” in numeric exprs.)
• Unary operators take 1 operand (e.g., −3);
binary operators take 2 operands (eg 3  4).
• Propositional or Boolean operators operate
on propositions (or their truth values)
instead of on numbers.
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Topic #1.0 – Propositional Logic: Operators
Module #1 - Logic
Some Popular Boolean Operators
Formal Name
Nickname Arity
Negation operator
NOT
Unary
¬
Conjunction operator
AND
Binary

Disjunction operator
OR
Binary

Exclusive-OR operator XOR
Binary

Implication operator
IMPLIES
Binary
Biconditional operator
IFF
Binary

↔
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Symbol
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
The Negation Operator
The unary negation operator “¬” (NOT)
transforms a prop. into its logical negation.
E.g. If p = “I have brown hair.”
then ¬p = “I do not have brown hair.”
p p
The truth table for NOT:
T F
T :≡ True; F :≡ False
F T
“:≡” means “is defined as”
Operand
column
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Result
column
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
The Conjunction Operator
The binary conjunction operator “” (AND)
combines two propositions to form their
ND
logical conjunction.
E.g. If p=“I will have salad for lunch.” and
q=“I will have steak for dinner.”, then
pq=“I will have salad for lunch and
I will have steak for dinner.”
Remember: “” points up like an “A”, and it means “ND”
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Conjunction Truth Table
Operand columns
• Note that a
p q
pq
conjunction
F F
F
p1  p2  …  pn
F T
F
of n propositions
T F
F
will have 2n rows
in its truth table.
T T
T
• Also: ¬ and  operations together are sufficient to express any Boolean truth table!
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
The Disjunction Operator
The binary disjunction operator “” (OR)
combines two propositions to form their
logical disjunction.
p=“My car has a bad engine.”

q=“My car has a bad carburetor.”
pq=“Either my car has a bad engine, or
the downwardmy car has a bad carburetor.” After
pointing “axe” of “”
Meaning is like “and/or” in English.
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splits the wood, you
can take 1 piece OR the
other, or both.
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Disjunction Truth Table
• Note that pq means
p q pq
that p is true, or q is
F F F
true, or both are true!
Note
F T T difference
• So, this operation is
T
F
T
from AND
also called inclusive or,
T
T
T
because it includes the
possibility that both p and q are true.
• “¬” and “” together are also universal.
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Nested Propositional Expressions
• Use parentheses to group sub-expressions:
“I just saw my old friend, and either he’s
grown or I’ve shrunk.” = f  (g  s)
– (f  g)  s would mean something different
– f  g  s would be ambiguous
• By convention, “¬” takes precedence over
both “” and “”.
– ¬s  f means (¬s)  f , not ¬ (s  f)
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
A Simple Exercise
Let p=“It rained last night”,
q=“The sprinklers came on last night,”
r=“The lawn was wet this morning.”
Translate each of the following into English:
¬p
= “It didn’t rain last night.”
lawn was wet this morning, and
r  ¬p
= “The
it didn’t rain last night.”
¬ r  p  q = “Either the lawn wasn’t wet this
morning, or it rained last night, or
the sprinklers came on last night.”
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
The Exclusive Or Operator
The binary exclusive-or operator “” (XOR)
combines two propositions to form their
logical “exclusive or” (exjunction?).
p = “I will earn an A in this course,”
q = “I will drop this course,”
p  q = “I will either earn an A in this course,
or I will drop it (but not both!)”
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Exclusive-Or Truth Table
• Note that pq means
p q pq
that p is true, or q is
F F F
true, but not both!
F T T
• This operation is
T
F
T
called exclusive or,
T
T
F
because it excludes the
possibility that both p and q are true.
• “¬” and “” together are not universal.
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Note
difference
from OR.
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Natural Language is Ambiguous
Note that English “or” can be ambiguous
regarding the “both” case! p q p "or" q
“Pat is a singer or
F F
F
Pat is a writer.” - 
F T
T
“Pat is a man or
T F
T
Pat is a woman.” - 
T T
?
Need context to disambiguate the meaning!
For this class, assume “or” means inclusive.
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Topic #1.0 – Propositional Logic: Operators
Module #1 - Logic
The Implication Operator
antecedent
consequent
The implication p  q states that p implies q.
I.e., If p is true, then q is true; but if p is not
true, then q could be either true or false.
E.g., let p = “You study hard.”
q = “You will get a good grade.”
p  q = “If you study hard, then you will get
a good grade.” (else, it could go either way)
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Implication Truth Table
• p  q is false only when
p q pq
p is true but q is not true.
F F
T
• p  q does not say
F T T
that p causes q!
T F
F
• p  q does not require
T T T
that p or q are ever true!
• E.g. “(1=0)  pigs can fly” is TRUE!
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The
only
False
case!
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Examples of Implications
• “If this lecture ends, then the sun will rise
tomorrow.” True or False?
• “If Tuesday is a day of the week, then I am
a penguin.” True or False?
• “If 1+1=6, then Bush is president.”
True or False?
• “If the moon is made of green cheese, then I
am richer than Bill Gates.” True or False?
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Module #1 - Logic
Why does this seem wrong?
• Consider a sentence like,
– “If I wear a red shirt tomorrow, then Osama bin Laden
will be captured!”
• In logic, we consider the sentence True so long as
either I don’t wear a red shirt, or Osama is caught.
• But, in normal English conversation, if I were to
make this claim, you would think that I was lying.
– Why this discrepancy between logic & language?
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Module #1 - Logic
Resolving the Discrepancy
• In English, a sentence “if p then q” usually really
implicitly means something like,
– “In all possible situations, if p then q.”
• That is, “For p to be true and q false is impossible.”
• Or, “I guarantee that no matter what, if p, then q.”
• This can be expressed in predicate logic as:
– “For all situations s, if p is true in situation s, then q is also
true in situation s”
– Formally, we could write: s, P(s) → Q(s)
• That sentence is logically False in our example,
because for me to wear a red shirt and for Osama to
stay free is a possible (even if not actual) situation.
– Natural language and logic then agree with each other.
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Topic #1.0 – Propositional Logic: Operators
Module #1 - Logic
English Phrases Meaning p  q
•
•
•
•
•
•
•
•
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“p implies q”
“if p, then q”
“if p, q”
“when p, q”
“whenever p, q”
“q if p”
“q when p”
“q whenever p”
•
•
•
•
•
“p only if q”
“p is sufficient for q”
“q is necessary for p”
“q follows from p”
“q is implied by p”
We will see some equivalent
logic expressions later.
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Converse, Inverse, Contrapositive
Some terminology, for an implication p  q:
• Its converse is:
q  p.
• Its inverse is:
¬p  ¬q.
• Its contrapositive: ¬q  ¬ p.
• One of these three has the same meaning
(same truth table) as p  q. Can you figure
out which?
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Topic #1.0 – Propositional Logic: Operators
Module #1 - Logic
How do we know for sure?
Proving the equivalence of p  q and its
contrapositive using truth tables:
p
F
F
T
T
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q
F
T
F
T
q
T
F
T
F
p
T
T
F
F
pq q p
T
T
T
T
F
F
T
T
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
The biconditional operator
The biconditional p  q states that p is true if and
only if (IFF) q is true.
p = “Bush wins the 2004 election.”
q = “Bush will be president for all of 2005.”
p  q = “If, and only if, Bush wins the 2004
election, Bush will be president for all of 2005.”
I’m still
here!
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Frank
2004
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Biconditional Truth Table
• p  q means that p and q
have the same truth value.
• Note this truth table is the
exact opposite of ’s!
Thus, p  q means ¬(p  q)
p
F
F
T
T
q pq
F T
T F
F F
T T
• p  q does not imply
that p and q are true, or cause each other.
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Boolean Operations Summary
• We have seen 1 unary operator (out of the 4
possible) and 5 binary operators (out of the
16 possible). Their truth tables are below.
p
F
F
T
T
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q
F
T
F
T
p pq pq pq pq pq
T F
F
F
T
T
T F
T
T
T
F
F F
T
T
F
F
F T
T
F
T
T
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Module #1 - Logic
Topic #1.0 – Propositional Logic: Operators
Some Alternative Notations
Name:
Propositional logic:
Boolean algebra:
C/C++/Java (wordwise):
C/C++/Java (bitwise):
not and or
  
p pq +
! && ||
~ & |
xor implies



!=
^
iff

==
Logic gates:
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